equations - Solving Equations Introductory example. Solve...

Solving Equations Introductory example . Solve the equation = x ( ) + x 2 15 ------- (i). We need to find all the real numbers x such the statement = x ( ) + x 2 15 is true. Putting the problem into words, we need to find a number such that the product of this number with a second number, that is obtained by adding 2 to the first number, is 15. Thinking just in terms of integers (whole numbers) we can quickly find the solution = x 3 by a "trial and error" approach. However, there is another solution. Rewriting the equation in the form = + x 2 2 x 15 by expanding the left side using distributivity of multiplication over addition. Then we can rewrite the equation in the form = + - x 2 2 x 15 0. so that the left side can be factored to give = ( )- x 3 ( ) + x 5 0 ------- (ii). At this stage we can use the fact that a product of two real numbers a and b is equal to zero exactly when either one or other of the two numbers a and b is equal to zero, that is, = a b 0 exactly when = a 0 or = b 0. ________________ This is equivalent to the following two facts taken together. • Multiplying any real number by 0 gives 0, that is, = 0. a 0 for any real number a . • Multiplying two non-zero numbers together gives a product that is not zero, that is, if ≠ a 0 and ≠ b 0, then ≠ a b 0. We apply this "zero product principle" to equation (ii) to see that = ( )- x 3 ( ) + x 5 0 exactly when either = - x 3 0 or = + x 5 0, which means the same as saying = x 3 or = x-5. Terminology used in connection with solving equations . If we introduce the concept of the solution set of an equation as the set of all solutions, then the solution set of equation (i) is { ,-5 3}, that is, the solution set is { x | x is a real number and = x ( ) + x 2 15 } = { ,-5 3}. In general, two equations are said to be equivalent exactly when they have the same solution set. The process of solving an equation (usually) involves writing down a sequence of equivalent equations, ending up with an equation from which the solutions can readily be obtained. The logical connective <=> which means "is equivalent to" may be used here. Thus the solution of equation (i) can be written down as follows. = x ( ) + x 2 15 <=> = + x 2 2 x 15 <=> = - - x 2 2 x 15 0 <=> = ( )- x 3 ( ) + x 5 0 <=> = x 3 or = x-5. _______ Each occurrence of <=> can be read as either "is equivalent to" or "which is equivalent to". One can then finish by stating that the solution set of the original equation is { , 3-5}. The symbol <=> is often omitted, however.

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Processes that produce an equivalent equation when applied to an equation involving a single variable or "unknown" x . • Adding a constant or an algebraic expression involving the variable x to both sides of an equation provided that the expression added can be evaluated for all real numbers. ( More generally, it is sufficient for the expression that is to be added to both sides to have a real number value for each of the

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